On Friday, October 4, 2013 10:47:19 PM UTC-5, Nam Nguyen wrote:> Two major theorems Godel's Incompleteness are:> - Incompleteness: Any formal system T that is consistent _and_ adequate> enough to describe the concept of the natural numbers, would have G(T)> as a statement that is true but not provable in T.

No it's not. An example of a formal system that is both complete and consistent is Peano's axioms.

The correct statement is that any axiomatization in FIRST ORDER logic powerful enough to embed the recursive functions is either incomplete or inconsistent. In the case of a first order axiomatization of number theory, the "incomplete" part means "incomplete" relative to the Peano Axioms.

Another, more direct, way of stating the result is this:

Peano's axioms are an essentially second order axiomatization.

or this:

There is no finite (or even recursively enumerable) system of first order axioms that completely encapsulates Peano's axioms

or more simply, still, this:

The axiom of induction (which is the one and only second order statement in Peano's axioms) cannot be equivalently decomposed into a finite or even recursively enumerable set of first order axioms.

or just this:

The axiom of induction has no equivalent formulation in first order logic.

> - Completeness: Any consistent formal system has to have a model.

Not so, either.

The correct statement is that there is an axiomatization of first order logic that is both complete and consistent.

If I recall correctly, the axiomatization of first order logic is, itself, in first order logic, otherwise the result wouldn't mean much.